J. Differential Equations 235 (2007) 376–396
www.elsevier.com/locate/jde
The problem of Dirichlet for anisotropic quasilinear
degenerate elliptic equations
Alkis S. Tersenov a,∗ , Aris S. Tersenov b
a University of Crete, Department of Mathematics, 71409 Heraklion-Crete, Greece
b University of Peloponnese, Department of Computer Science and Technology, 22100 Tripoli, Greece
Received 11 January 2006; revised 8 October 2006
Available online 24 January 2007
Abstract
We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori
estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient
conditions guaranteeing the solvability of the problem are formulated. The results are new even in the
semilinear case when the principal part is the Laplace operator.
© 2007 Elsevier Inc. All rights reserved.
MSC: 35J25; 35J70; 35B45
Keywords: Degenerate elliptic equations; Semilinear elliptic equations; A priori estimates
0. Introduction and main results
In the present paper we consider the following quasilinear degenerate elliptic equation
−
n
μi |uxi |pi uxi x = c(x)g(u) + f (x)
i
in Ω ⊂ Rn ,
(0.1)
i=1
coupled with homogeneous Dirichlet boundary condition
u = 0 on ∂Ω.
* Corresponding author.
E-mail addresses: [email protected] (Al.S. Tersenov), [email protected] (Ar.S. Tersenov).
0022-0396/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jde.2007.01.009
(0.2)
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
377
Here constants μi > 0 and pi 0. Without loss of generality we assume that
Ω ⊂ {x: −li xi li , i = 1, . . . , n}.
Concerning the function g we suppose that
g(z) > 0 if z > 0 and g(z) g(C)
g(0) = 0,
for |z| C,
(0.3)
where C is an arbitrary positive constant. For example, functions g(u) = ln(|u| + 1) and
g(u) = uq with arbitrary q 0 satisfy (0.3). In the case that g(u) = uq is defined only for u 0
one can take its odd or even continuation of the form g(u) = |u|q−1 u, g(u) = |u|q . On the other
hand, it is obvious that (0.3) does not restrict us only to odd or even functions. For example
g(u) = eu − 1, which is neither odd nor even, satisfies condition (0.3).
For the equation
−
n
μi |uxi |pi uxi x = f (x)
i
in Ω ⊂ Rn
i=1
with condition (0.2) the existence of the unique generalized solution follows from [8]. In [8]
the initial boundary value problem for the related parabolic equation was considered, but the
method proposed there can be easily applied to the elliptic case. From [8] it follows that if
f ∈ W −1,p0 (Ω) (where p0 = max1in {pi }, 1/pi + 1/pi = 1), then there exists a unique generalized solution such that u ∈ L2 (Ω) and uxi ∈ Lpi (Ω).
The existence and nonexistence of positive solutions for equation
−
n
μi |uxi |pi uxi x = λuq ,
λ > 0 is constant,
i
i=1
coupled with boundary condition (0.2) was considered in [3]. The existence result was proved in
the subcritical case and nonexistence result in the at least critical case.
In [7] the regularity question for the equation
n
μi |uxi |pi uxi x = 0,
i
i=1
was considered. It was proved that each component of the gradient is bounded in L∞ norm under
the assumption that the solution is bounded.
Our goal is to obtain sufficient conditions for the solvability of problem (0.1), (0.2). The
results obtained in the present paper are new even for the semilinear case, i.e. for p1 = p2 =
· · · = pn = 0.
Assume that there exists a positive constant M such that
c0 g(M) + f0
3l 2 + 2l p+1
2
< μ(p + 1)M p+1 .
Here p = pi0 = max{p1 , . . . , pn }, μ = μi0 , l = li0 , c0 = supΩ |c(x)| and f0 = supΩ |f (x)|.
Below we will give several examples concerning this condition.
(0.4)
378
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Remark 1. Instead of (0.4) we can take one of the following two assumptions.
1. Suppose that there exists a positive constant M such that
3l˜2 + 2l˜ p̃+1
< μ̃(p̃ + 1)M p̃+1 .
c0 g(M) + f0
2
(0.41 )
Here l˜ = li1 = min{l1 , . . . , ln }, μ̃ = μi1 , p̃ = pi1 .
2. Suppose that there exists a positive constant M such that
3lˆ2 + 2lˆ p̂+1
c0 g(M) + f0
< μ̂(p̂ + 1)M p̂+1 .
2
(0.42 )
Here μ̂ = μi2 = max{μ1 , . . . , μn }, lˆ = li2 , p̂ = pi2 .
Definition 1. We say that function u(x) is a generalized solution of problem (0.1), (0.2) if u(x) ∈
W 1,∞ (Ω), u(x) = 0 for x ∈ ∂Ω and
n
μi |uxi | uxi φxi (x) dx =
pi
Ω i=1
◦
c(x)g(u) + f (x) φ(x) dx ∀φ ∈ W 1,r (Ω), 1 r < +∞.
Ω
Theorem 1.
(i) Suppose that c(x) and f (x) are bounded in Ω̄, g(u) is a Hölder continuous function on
[−M, M] and conditions (0.3), (0.4) are fulfilled. If the domain Ω ⊂ Rn is strictly convex,
then there exists a generalized solution of problem (0.1), (0.2) such that
uL∞ (Ω) M0
and uxi L∞ (Ω) (1 + 2li )
Φ0
μi (1 + pi )
1
pi +1
,
i = 1, . . . , n,
where M0 = inf{M: M satisfies (0.4)} and
Φ0 =
max
c(x)g(u) + f (x).
Ω̄×[−M0 ,M0 ]
(ii) If in addition c(x) 0 and g(u) is a nondecreasing function then the solution is unique.
Example 1. Consider the following equation
−
n
μi |uxi |uxi x = c(x)u4 + f (x)
i
in Ω.
i=1
Here p = 1 and g(u) = u4 . Let c0 > 0, then inequality (0.4) takes the following form
c0 M 4 −
(3l 2
8μ
M 2 + f0 < 0
+ 2l)2
(0.5)
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
379
or for M̄ = M 2
c0 M̄ 2 −
8μ
M̄ + f0 < 0.
+ 2l)2
(0.6)
16μ2
.
c0 (3l 2 + 2l)4
(0.7)
(3l 2
Obviously, M̄ > 0 satisfying (0.6) exists if
f0 Thus if the function f (x) satisfies condition (0.7), then Theorem 1 guarantees the existence of a
generalized solution of problem (0.5), (0.2) satisfying inequalities
uL∞ (Ω) M0
1
c0 M04 + f0 2
and uxi L∞ (Ω) (1 + 2li )
,
2μi
i = 1, . . . , n,
with
M0 =
1 1
4μ
16μ2
f0 2 2
− 2
−
.
c0
c0 (3l 2 + 2l)2
c0 (3l 2 + 2l)4
Example 2. If g(u) = uq (or g(u) = |u|q−1 u or g(u) = |u|q ) and p + 1 > q then for arbitrary
bounded f (x) one can find a positive M satisfying condition (0.4) and as a consequence obtain
the existence of a generalized solution by Theorem 1.
Example 3. If c0 = 0, then (as in the previous case) one can always find positive M satisfying
(0.4) and obtain the existence of a generalized solution for any bounded f (x). In this case
1
p+1
f0
3l 2 + 2l
uL∞ (Ω) M0 =
,
2
μ(p + 1)
1
pi +1
f0
, i = 1, . . . , n.
uxi L∞ (Ω) (1 + 2li )
μi (1 + pi )
Consider now the semilinear equation (pi = 0 for all i). For simplicity suppose that μi = μ
for all i:
−μu = c(x)g(u) + f (x)
in Ω ⊂ Rn .
(0.8)
In this case the use of (0.41 ) is appropriate, for pi = 0 it takes the form
c0 g(M) + f0
3l˜2 + 2l˜
2
< μM.
(0.9)
In [9] it was shown that problem (0.8), (0.2) with f ≡ 0 and c(x) ≡ const has a nontrivial solution. It is natural to expect that problem (0.8), (0.2) with arbitrary f may have no solution. Let us
380
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
mention here that many papers have recently appeared (see [1,2,10] and the references there)
where for the problem
−u = |u|q−1 u + f (x)
u = 0 on ∂Ω,
in Ω,
different sufficient conditions on existence are formulated.
Theorem 2.
(i) Suppose that c(x), f (x) ∈ C γ (Ω̄), g(u) ∈ C γ ([−M, M]), ∂Ω ∈ C 2+γ , γ ∈ (0, 1). If (0.9)
is fulfilled then there exists a classical solution of problem (0.8), (0.2) u(x) ∈ C 2+γ (Ω̄) such
that
max |u| M0 ,
Ω
where M0 = inf{M: M satisfies (0.9)}.
(ii) If in addition c(x) 0 and g(u) is a nondecreasing function then the solution is unique.
Remark 2. In order to prove Theorem 1 we need to obtain a priori estimates for u and ∇u. To
prove the estimate for ∇u we will use the convexity of the domain. In order to prove Theorem 2
we need to obtain a priori estimates only for u where we do not need the convexity of the domain.
If the domain in Theorem 2 is strictly convex then we additionally have
max |uxi | (1 + 2li )
Ω
Φ0
,
μ
i = 1, . . . , n.
Example 4. Consider the equation
−μu = λu2 + f (x),
where λ is constant.
Condition (0.9) takes the form
|λ|M 2 −
2μ
3l˜2
+ 2l˜
M + f0 < 0.
From Theorem 2 we have that there exists a classical solution if
f0 μ2
,
˜2
|λ|(3l˜2 + 2l)
and in this case
1
μ2
μ
f0 2
−
−
.
max |u| M0 =
˜
˜ 2 |λ|
Ω
|λ|(3l˜2 + 2l)
λ2 (3l˜2 + 2l)
If the domain is strictly convex then in addition
max |uxi | (1 + 2li )
Ω
|λ|M02 + f0
,
μ
i = 1, . . . , n.
(0.10)
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
381
Example 5. Consider
−μu = λeu .
(0.11)
Here g(u) = eu − 1 and f = λ. Condition (0.9) takes the form: there exists M > 0 such that
eM − CM < 0,
where
C=
2μ
.
˜
|λ|(3l˜2 + 2l)
This fact is equivalent to say that the minimum of the function ex − Cx must be negative, that is
2μ
> e.
˜
|λ|(3l˜2 + 2l)
(0.12)
From Theorem 2 we have that there exists a classical solution of (0.11), (0.2) if (0.12) is fulfilled
and for this solution we have
max |u| M0 = ln C.
(0.13)
Ω
If the domain is strictly convex then in addition
max |uxi | (1 + 2li )
Ω
2(1 + 2li )
|λ|eln C
=
,
μ
3l˜2 + 2l˜
i = 1, . . . , n.
(0.14)
Recall that in [4] for the problem
−u = eu
in Ω ⊂ Rn ,
u|∂Ω = 0
(0.15)
it was shown that there exists a positive number κ depending on the dimension n such that if
the diameter of Ω is less than κ there exists (at least one) solution of problem (0.15). From
Example 5 it follows that there exists (at least one) solution of problem (0.15) if the size of the
domain Ω at least in one direction is small enough (independently of the dimension) namely
2
3l˜ 2 + 2l˜ < .
e
Finally let us mention that similarly to problem (0.11), (0.2) we can show that if (0.12) is
fulfilled then there exists a classical solution of the problem
μu = λeu
in Ω,
u|∂Ω = 0
satisfying inequality (0.13) (and, if Ω is strictly convex, inequality (0.14)). Moreover from Theorem 2 it follows that the solution is unique.
The paper consists of two sections. In the first section we obtain the a priori estimate for the
regularized problem and in the second based on these a priori estimates we prove Theorems 1, 2.
382
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
1. A priori estimates for the regularized problem
Let us start from problem (0.1), (0.2). Consider the regularized equation
−
n
p /α μi uαxi + ε i uxi x = cε (x)gM (u) + fε (x).
i
(1.1)
i=1
Here the constant α ∈ (0, 1) is such that (uαxi )pi /α = |uxi |pi (for example α = 2/3); constant ε > 0, cε (x), fε (x) are Hölder continuous functions such that
cε (x) → c(x)
and fε (x) → f (x)
in L∞ norm, when ε → 0,
where without loss of generality we assume that
maxcε (x) = c0 ,
maxfε (x) = f0 ,
maxcε (x)g(u) + fε = Φ0 ;
and finally
g(z),
gM (z) =
g(M),
g(−M),
for |z| M,
for z > M,
for z < −M.
(1.2)
Obviously from (0.3) we have −g(M) gM (u) g(M).
The first step is to obtain the estimate |u| M for a solution of problem (1.1), (0.2). After this
in (1.1) instead of gM (u) we can take g(u) (due to (1.2)).
Lemma 1. If (0.3) and (0.4) are fulfilled, then for any classical solution of problem (1.1), (0.2)
the following estimate is valid
u(x) M.
Proof. Without loss of generality we assume that i0 = 1, i.e. p = p1 , l = l1 , μ = μ1 . Rewrite
Eq. (1.1) in nondivergent form
−
n
aiε (uxi )uxi xi = cε (x)gM (u) + fε (x),
i=1
where
pi −1 aiε (z) = μi zα + ε α
(pi + 1)zα + ε .
Define the function h(x1 ):
l 2 − x12
+ (1 + l)(l + x1 ) ,
h(x1 ) = M̃
2
(1.3)
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
383
where
M̃ =
2M
.
+ 2l
3l 2
Obviously for
Lu ≡ −
n
aiε (uxi )uxi xi
i=1
we have
Lu = cε (x)gM (u) + fε (x)
(1.4)
and
Lh = −
n
aiε (hxi )hxi xi = −a1ε (hx1 )hx1 x1
i=1
p −1 (p + 1)h α + ε M̃ μ(p + 1)M̃ p+1 .
= μ h α + ε α
(1.5)
Here we use the fact that h (x1 ) M̃, h (x1 ) = −M̃ and α < 1. One can easily see that if
p
α ∈ (0, 1) then for any p 0 the expression μ(h α + ε) α −1 ((p + 1)h α + ε)M̃ is a nondecreasing
with respect to ε function. For the function
v(x) ≡ u(x) − h(x1 )
we have
Lu − Lh = −
n
aiε (uxi )uxi xi +
i=1
=−
n
n
aiε (hxi )hxi xi
i=1
aiε (uxi )vxi xi + a1ε (hx1 ) − a1ε (ux1 ) hx1 x1 .
i=1
On the other hand, due to (1.4), (1.5) we have
Lu − Lh = cε (x)gM (u) + fε (x) − Lh cε (x)gM (u) + fε (x) − μ(p + 1)M̃ p+1 .
Hence
−
n
aiε (uxi )vxi xi a1ε (ux1 ) − a1ε (hx1 ) hx1 x1
i=1
+ cε (x)gM (u) + fε (x) − μ(p + 1)M̃ p+1 .
(1.6)
384
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Suppose that at the point N ∈ Ω̄ \ ∂Ω the function v(x) attains its positive maximum. At this
point we have v > 0 and vxi = 0 or u > h 0 and ux1 = h M̃, uxi = 0 for i = 2, . . . , n
(in particular a1ε (ux1 ) − a1ε (hx1 ) = 0). Thus
−
n
aiε (uxi )vxi xi cε (x)gM (u) + fε (x) − μ(p + 1)M̃ p+1 N
N
i=1
maxcε (x)g(M) + maxfε (x) − μ(p + 1)M̃ p+1
p+1
2M
= c0 g(M) + f0 − μ(p + 1)
.
3l 2 + 2l
(1.7)
Here we use the fact that for positive u we have 0 gM (u) g(M). Hence due to (0.4)
−
n
aiε (uxi )vxi xi < 0.
N
i=1
This contradicts the assumption that v(x) attains its positive maximum at N . Due to the homogeneous boundary conditions, on ∂Ω we have v = −h 0. Taking into account that v(x) cannot
attain its positive maximum in Ω̄ \ ∂Ω we conclude that
v(x) 0 or
u(x) h(x1 )
in Ω̄.
Now let us obtain the estimate from the below.
For the function w(x) ≡ u(x) + h(x1 ) we have
Lu + Lh = −
n
i=1
=−
n
aiε (uxi )uxi xi −
n
aiε (hxi )hxi xi
i=1
aiε (uxi )wxi xi − a1ε (hx1 ) − a1ε (ux1 ) hx1 x1 .
i=1
On the other hand,
Lu + Lh = cε (x)gM (u) + fε (x) + Lh cε (x)gM (u) + fε (x) + μ(p + 1)M̃ p+1 .
Thus
−
n
aiε (uxi )wxi xi a1ε (hx1 ) − a1ε (ux1 ) hx1 x1 + cε (x)gM (u) + fε (x) + μ(p + 1)M̃ p+1 .
i=1
Suppose that at the point N1 ∈ Ω̄ \ ∂Ω the function w(x) attains its negative minimum. At
this point we have w < 0 and wxi = 0 or u < −h 0 and ux1 = −hx1 −M̃, uxi = 0 for
i = 2, . . . , n. Therefore (due to the fact that aiε (z) = aiε (−z))
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
−
n
aiε (uxi )wxi xi i=1
cε (x)gM (u) + fε (x) + μ(p + 1)M̃ p+1 N
N1
−c0 g(M) − f0 + μ(p + 1)
2M
2
3l + 2l
385
1
p+1
.
(1.8)
Here we use the inequality
cε (N1 )gM u(N1 ) −c0 g(M).
If cε (N1 ) 0, then the last inequality follows from the fact that gM (u) −g(M). If cε (N1 ) < 0,
then the inequality follows from the fact that gM (u) g(M). Hence from (1.8) (due to (0.4)) we
obtain
−
n
i=1
aiε (uxi )wxi xi N1
> 0.
This contradicts the assumption that w(x) attains its (negative) minimum at N1 .
Due to the homogeneous boundary condition, on ∂Ω we have w = h 0. Taking into account
that w(x) cannot attain its negative minimum in Ω̄ \ ∂Ω we conclude that
w(x) 0 or u(x) −h(x1 )
in Ω̄.
Consequently
−h(x1 ) u(x) h(x1 ).
(1.9)
Now taking h̃(x1 ) ≡ h(−x1 ) instead of h(x1 ) we obtain
−h̃(x1 ) u(x) h̃(x1 ).
(1.10)
This estimate can be easily established in the same way as (1.9) because h̃ α M̃ α and
h̃ = −M̃. The first inequality (h̃ α M̃ α ) follows from −h̃ M̃ 0 due to the choice of α
(α = 2/3).
From (1.9) and (1.10) we conclude that
2
u(x) h(0) = h̃(0) = 3l + 2l M̃ = M.
2
2
Remark 3. Represent M for a sufficiently small as M = M0 + , where M0 = inf{M:
M satisfies (0.4)}. Passing to the limit when → 0 we obtain the estimate
u(x) M0 .
Let us turn to the estimate of the derivatives. First in Lemma 2 we will obtain the auxiliary
result which actually is the boundary gradient estimate. Then in Lemma 3 we will obtain the
386
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
global gradient estimate. We additionally suppose now that Ω is strictly convex and that the
parts of ∂Ω lying in the half spaces xi 0 and xi 0 can be expressed as
xi = Fi
and xi = Gi ,
i = 1, . . . , n,
respectively. Here the functions Fi and Gi depend on all variables except of xi . Due to the
convexity we have
Fkxi xi 0,
Gkxi xi 0,
k = 1, 2, . . . , n, i = 1, 2, . . . , n.
Define the function hk (τ ) by the following
hk (τ ) = −Ck
τ2 + Ck (1 + 2lk ) + τ,
2
τ ∈ [0, 2lk ],
(1.11)
where
Ck =
Φ0
μk (pk + 1)
1
pk +1
.
Recall that Φ0 = maxΩ̄×[−M,M] |cε (x)g(u) + fε (x)|. Obviously
hk = −Ck ,
hk Ck + > Ck .
Lemma 2. If conditions (0.3), (0.4) are fulfilled and Ω is strictly convex then for any classical
solution of problem (1.1), (0.2) the following estimates are valid
u(x) hk (Gk − xk ),
u(x) hk (xk − Fk ),
k = 1, . . . , n, in Ω̄.
Proof. We will prove these estimates for k = 1, the other cases can be considered similarly. Let
us start from the first inequality. Introduce the function
v(x) ≡ u(x) − h1 (ζ ),
where ζ = G1 (x2 , x3 , . . . , xn ) − x1 .
We have
Lu(x) ≡ −
n
aiε (uxi )uxi xi = cε (x)g(u) + fε (x),
i=1
Lh1 (ζ ) = −
n
aiε h1xi (ζ ) h1xi xi (ζ )
i=1
n
aiε h1 (ζ )G1xi h1 (ζ )G21xi + h1 (ζ )G1xi xi .
= −a1ε h1 (ζ ) h1 (ζ ) −
i=2
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
387
Due to the convexity of Ω we have G1xi xi 0 and taking into account that h1 0, h1 0 we
conclude that
h1 (ζ )G21xi + h1 (ζ )G1xi xi 0
and hence
Lh1 (ζ ) −a1ε h1 (ζ ) h1 (ζ ) = C1 a1ε h1 (ζ ) .
(1.12)
Lu(x) − Lh1 (ζ ) cε (x)g(u) + fε (x) − C1 a1ε h1 (ζ ) .
(1.13)
Thus
On the other hand,
Lu(x) − Lh1 (ζ ) = −
n
aiε (uxi )vxi xi −
i=1
n
aiε (uxi ) − aiε h1xi (ζ ) h1xi xi (ζ ).
(1.14)
i=1
Hence, from (1.13), (1.14) we obtain
−
n
aiε (uxi )vxi xi
n
aiε (uxi ) − aiε h1xi (ζ ) h1xi xi (ζ )
i=1
i=1
+ cε (x)g(u) + fε (x) − C1 a1ε h1 (ζ ) .
(1.15)
Suppose that at the point N ∈ Ω̄ \ ∂Ω the function v(x) attains its maximum. At this point we
have vxi = 0 or uxi (x) = h1xi (ζ ) (in particular aiε (uxi ) − aiε (h1xi (ζ )) = 0, i = 1, . . . , n) and
hence
−
n
aiε (uxi )vxi xi cε (x)g(u) + fε (x) − C1 a1ε h1 (ζ ) N
N
i=1
p +1
< cε (N )g u(N) + fε (N ) − μ1 (p1 + 1)C1 1 0.
This contradicts the assumption that v(x) attains its maximum at the internal point of the domain Ω. Due to the fact that v = −h1 0 on ∂Ω we conclude that
v(x) 0 or
u(x) h1 (G1 − x1 )
in Ω̄.
Next we obtain a lower bound. Introduce the function w(x) ≡ u(x)+h1 (ζ ). Similarly to (1.13)
and (1.14) we obtain
Lu(x) + Lh1 (ζ ) cε (x)g(u) + fε (x) + C1 a1ε h1 (ζ )
and
Lu(x) + Lh1 (ζ ) = −
n
i=1
aiε (uxi )wxi xi +
n
aiε (uxi ) − aiε h1xi (ζ ) h1xi xi (ζ ).
i=1
388
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Hence
−
n
aiε (uxi )wxi xi −
i=1
n
aiε (uxi ) − aiε h1xi (ζ ) h1xi xi (ζ )
i=1
+ cε (x)g(u) + fε (x) + C1 a1ε h1 (ζ ) .
(1.16)
Suppose that at the point N1 ∈ Ω̄ \ ∂Ω the function w(x) attains its minimum. At this point we
have wxi = 0 or uxi (x) = h1xi (ζ ) (in particular aiε (uxi ) − aiε (h1xi (ζ )) = 0, i = 1, . . . , n) and
hence
−
n
aiε (uxi )wxi xi i=1
N1
cε (x)g(u) + fε (x) + C1 a1ε h (ζ ) N
1
p +1
> cε (N )g u(N) + fε (N ) + μ1 (p1 + 1)C1 1 0.
This contradicts the assumption that w(x) attains its minimum at the internal point of the domain Ω. Due to the fact that w = h1 0 on ∂Ω we conclude that
w(x) 0 or
u(x) −h1 (G1 − x1 )
in Ω̄.
Thus the estimate |u(x)| h1 (G1 − x1 ) in Ω̄ is proved.
Now introduce functions ṽ(x) ≡ u(x) − h1 (η) and w̃(x) ≡ u(x) + h1 (η) where η = x1 −
F1 (x2 , x3 , . . . , xn ). Similarly to (1.12) we obtain
Lh1 (η) = −
n
aiε h1xi (η) h1xi xi (η)
i=1
n
2
aiε h1 (η)F1xi h1 (η)F1x
− h1 (η)F1xi xi .
= −a1ε h1 (η) h1 (η) −
i
i=2
Due to the convexity of Ω we have F1xi xi 0 and taking into account that h1 0, h1 0 we
conclude that
2
− h1 (ζ )F1xi xi 0
h1 (ζ )F1x
i
and hence
Lh1 (η) −a1ε h1 (η) h1 (η) = C1 a1ε h1 (η) .
Furthermore similarly to (1.15) and (1.16) we obtain
−
n
i=1
aiε (uxi )ṽxi xi n
aiε (uxi ) − aiε h1xi (η) h1xi xi (η)
i=1
+ cε (x)g(u) + fε (x) − C1 a1ε h1 (η)
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
389
and
−
n
aiε (uxi )w̃xi xi −
i=1
n
aiε (uxi ) − aiε h1xi (η) h1xi xi (η)
i=1
+ cε (x)g(u) + fε (x) + C1 a1ε h1 (η) .
Now in the same manner as in the previous case we obtain the estimate |u(x)| h1 (x1 − F1 )
in Ω̄.
Lemma is proved. 2
Lemma 3. If conditions (0.3), (0.4) are fulfilled and Ω is strictly convex, then for any classical
solution of problem (1.1), (0.2) the following estimates are valid
ux (x) (1 + 2li )
i
Φ0
μi (1 + pi )
1
pi +1
,
i = 1, . . . , n.
Proof. We will prove the estimate for i = 1, for i = 2, . . . , n the proof is similar. Consider the
equations
n
−a1ε ux1 (x) ux1 x1 (x) −
aiε uxi (x) uxi xi (x) = cε (x)g u(x) + fε (x),
(1.17)
i=2
n
aiε uxi (x̃) uxi xi (x̃) = cε (x̃)g u(x̃) + fε (x̃),
−a1ε uξ (x̃) uξ ξ (x̃) −
(1.18)
i=2
where x = (x1 , x2 , . . . , xn ) and x̃ = (ξ, x2 , . . . , xn ). Subtracting Eq. (1.18) from (1.17) for
v(ξ, x) ≡ u(x) − u(x̃)
we obtain
n
−a1ε ux1 (x) vx1 x1 − a1ε uξ (x̃) vξ ξ −
aiε uxi (x) uxi xi (x) − aiε uxi (x̃) uxi xi (x̃)
i=2
= cε (x)g u(x) + fε (x) − cε (x̃)g u(x̃) − fε (x̃).
Rewrite this equation in the following form
n
−a1ε ux1 (x) vx1 x1 − a1ε uξ (x̃) vξ ξ −
aiε uxi (x) vxi xi
i=2
n
aiε uxi (x) − aiε uxi (x̃) uxi xi (x̃)
=
i=2
+ cε (x)g u(x) + fε (x) − cε (x̃)g u(x̃) − fε (x̃).
(1.19)
390
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Consider (1.19) in the domain
Q = (ξ, x): ξ ∈ (F1 , G1 ), x1 ∈ (F1 , G1 ), x1 > ξ, (x2 , . . . , xn ) ∈ Ω1 ,
where Ω1 is a projection of Ω on the hyperplane x1 = 0. For
w(ξ, x) = v(ξ, x) − h1 (x1 − ξ )
we have
n
−a1ε ux1 (x) wx1 x1 − a1ε uξ (x̃) wξ ξ −
aiε uxi (x) wxi xi
i=2
=
n
aiε uxi (x) − aiε uxi (x̃) uxi xi (x̃)
i=2
+ cε (x)g u(x) + fε (x) − cε (x̃)g u(x̃) − fε (x̃) + a1ε ux1 (x) + a1ε uξ (x̃) h1
n
aiε uxi (x) − aiε uxi (x̃) uxi xi (x̃) + 2Φ0 − C1 a1ε ux1 (x) + a1ε uξ (x̃) .
i=2
(1.20)
Suppose that at the point N ∈ Q̄ \ ∂Q the function w(ξ, x) attains its maximum. At this point we
have wξ = wxi = 0, i = 1, . . . , n, or
ux1 (x) = uξ (x̃) = h1
and uxi (x) = uxi (x̃)
for i = 2, . . . , n.
Hence from (1.20) we have
n
−a1ε ux1 (x) wx1 x1 − a1ε uξ (x̃) wξ ξ −
aiε uxi (x) wxi xi i=2
N
p
2 Φ0 − C1 μ1 h1 1 (p1 + 1) N < 2(Φ0 − Φ0 ) = 0.
This contradicts the assumption that w(ξ, x) attains its maximum at the internal point of the
domain Q.
Now consider w(ξ, x) on ∂Q. The boundary of Q consists of three parts (recall that x1 > ξ ):
(1) x1 = ξ ;
(2) ξ = F1 , x1 ∈ [F1 , G1 ], x2 , . . . , xn ∈ Ω̄1 ;
(3) x1 = G1 , ξ ∈ [F1 , G1 ], x2 , . . . , xn ∈ Ω̄1 .
On the first part we obviously have w = −h1 (0) = 0. On the second and the third parts, due
to Lemma 2, we have respectively
w = u(x) − h1 (x1 − F1 ) 0
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
391
and
w = −u(x̃) − h1 (G1 − ξ ) 0.
Consequently w(ξ, x) 0 in Q̄, which means
u(x) − u(x̃) h1 (x1 − ξ )
in Q̄.
Similarly, taking the function ṽ ≡ u(x̃) − u(x) instead of v, we obtain v −h1 (x1 − ξ ) in Q̄.
By the symmetry of the variables x1 and ξ , we consider the case ξ > x1 in the same way.
As a result we obtain that for x1 ∈ [F1 , G1 ], ξ ∈ [F1 , G1 ], (x2 , . . . , xn ) ∈ Ω1 , |x1 − ξ | > 0 the
following inequality holds:
|u(x) − u(x̃)| h1 (|x1 − ξ |) − h1 (0)
,
|x1 − ξ |
|x1 − ξ |
which in turn implies the estimate |ux1 (x)| h1 (0) = (1 + 2l1 )C1 + . Passing to the limit when
→ 0 we conclude that
ux (x) (1 + 2l1 )
1
The lemma is proved.
Φ0
μ1 (p1 + 1)
1
p1 +1
.
2
Remark 4. When proving Lemma 3 we use the idea of S.N. Kruzhkov [6] of introducing a new
spatial variable for the one-dimensional quasilinear parabolic equations (see also [12] and the
references there). The extension of this method to a class of multidimensional elliptic equations
in convex domains was presented in [11,13]. In [11] (as well as in [6]) the right-hand side must
vanish at the points where the principal part becomes zero. Of course Eq. (0.1) does not satisfy
such restrictions. In [13] (as well as in [12]) it was shown that the a priori gradient estimate for
the degenerated equation can be established under specific restrictions on the right-hand side
which in our case look like
c(x)g(u) + f (x) − c(x )g(v) + f (x ) 0 for u > v,
where x = (x1 , x2 , x3 , . . . , xn ) with x1 > x1 , if we need the estimate of ux1 ; x = (x1 , x2 , x3 ,
. . . , xn ) with x2 > x2 , if we need the estimate of ux2 and so on. In the present paper we have
succeeded to obtain the needed estimate for (0.1) with arbitrary c(x)g(u) + f (x) due to the
specific form of the principal part. Note that in [6,11–13] the existence of classical solutions is
proved.
Let us turn now to problem (0.8), (0.2).
Consider the regularized equation
−μu = c(x)gM (u) + f (x).
(1.21)
Recall that here we suppose that c and f are Hölder continuous functions. The proofs of the
following two lemmas are similar to the proofs of Lemmas 1 and 3.
392
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Lemma 4. If (0.3) and (0.4) are fulfilled, then for any classical solution of problem (1.21), (0.2)
the following estimate is valid
u(x) M.
Lemma 5. If conditions (0.3), (0.4) are fulfilled and Ω is strictly convex, then for any classical
solution of problem (1.21), (0.2) the following estimates are valid
ux (x) (1 + 2li )
i
Φ0
μi (1 + pi )
1
pi +1
,
i = 1, . . . , n.
2. Existence and uniqueness
Proof of Theorem 1. Consider equation
−
n
p /α
μi uαεxi + ε i uεxi x = cε (x)g(uε ) + fε (x).
(2.1)
i
i=1
The classical solvability of problem (2.1), (0.2) follows from [5].
Our goal is to pass to the limit (ε → 0) in (2.1) based on the a priori estimates obtained in
previous section. Due to Lemmas 1 and 3 there exists a subsequence which we denote again by
uε such that
uε (x) → u(x)
uniformly in C0 norm
(2.2)
and
uεxi (x) → uxi (x)
∗ weakly in L∞ (Ω),
i = 1, 2, . . . , n.
(2.3)
From (2.2) it immediately follows that
fε ≡ cε (x)g(uε ) + fε (x) → f ≡ c(x)g(u) + f (x)
strongly in L∞ norm.
◦
Define Aε (uε ) and A(u) elements from W −1,s (Ω) (linear functionals on W 1,r (Ω),
by the following
n
Aε (uε ), v =
+
1
s
= 1)
◦
p /α
μi uαεxi + ε i uεxi vxi dx ∀v ∈ W 1,r (Ω),
i=1 Ω
n
A(u), v =
1
r
◦
μi |uxi |pi uxi vxi dx ∀v ∈ W 1,r (Ω).
i=1 Ω
From Lemma 3 it follows that μi (uαεxi + ε)pi /α uεxi is bounded in L∞ (Ω) and hence in Ls (Ω)
for any s. Thus
Aε (uε ) → χ
weakly in W −1,s (Ω).
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
393
Our goal is to prove that
χ = A(u).
One can easily see by direct calculations that
A(uε ) − A(v), uε − v 0.
Hence
Aε (uε ) − A(v), uε − v Aε (uε ) − A(uε ), uε − v .
(2.4)
Rewrite (2.4) as following
Aε (uε ), uε − Aε (uε ), v − A(v), uε − v Aε (uε ) − A(uε ), uε − v .
(2.5)
Multiplying (2.1) by uε and then integrating by part we obtain
n
Aε (uε ), uε =
p /α
μi uαεxi + ε i u2εxi dx =
i=1 Ω
fε uε dx ≡ (fε , uε ).
Ω
Hence from (2.5) it follows that
(fε , uε ) − Aε (uε ), v − A(v), uε − v Aε (uε ) − A(uε ), uε − v .
(2.6)
Passing to the limit when ε → 0 we obtain (see Remark 5 below)
(f, u) − χ, v − A(v), u − v 0.
(2.7)
Now multiplying (2.1) by u and integrating by parts we have
n
Aε (uε ), u =
p /α
μi uαεxi + ε i uεxi uxi dx =
i=1 Ω
fε u dx = (fε , u).
Ω
Passing to the limit when ε → 0 we obtain
χ, u = (f, u).
Substituting this in (2.7) we have
χ, u − χ, v − A(v), u − v 0 or χ − A(v), u − v 0.
◦
(2.8)
Select v ≡ u − λw, where λ is a positive constant and w ∈ W 1,∞ (Ω). From (2.8) we conclude
that
λ χ − A(u − λw), w 0 or
χ − A(u − λw), w 0.
394
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
Passing to the limit when λ → 0 (this is possible due to the Lebesgue theorem) we obtain
◦
χ − A(u), w 0 ∀w ∈ W 1,∞ (Ω).
Hence the functional χ − A(u) is zero, i.e.
χ = A(u).
Thus we can pass to the limit when ε → 0 in
Aε (uε ), φ = cε g(uε ) + f, φ
(2.9)
to obtain
A(u), φ = cg(u) + f, φ .
(2.10)
Obviously (2.9) and (2.10) are equivalent to
n
p /α
μi uαεxi + ε i uεxi φxi dx =
Ω i=1
cε (x)g(uε ) + fε (x) φ dx
Ω
and
n
c(x)g(u) + f (x) φ dx,
μi |uxi | uxi φxi dx =
pi
Ω i=1
Ω
respectively.
The existence is proved.
Let us pass to the uniqueness. Suppose that there exist two solutions u1 and u2 . We have
A(u1 ) − A(u2 ), u1 − u2 = c(x)g(u1 ) − c(x)g(u2 ), u1 − u2 .
(2.11)
The left-hand side of (2.11) is nonnegative, hence
c(x) g(u1 ) − g(u2 ) (u1 − u2 ) dx 0.
Ω
Due to the assumptions of Theorem 1 concerning the uniqueness, the last inequality takes place
only if u1 − u2 ≡ 0.
Theorem 1 is proved. 2
Remark 5. When passing to the limit in (2.6) we use the fact that μi (uαεxi + ε)pi /α uεxi and
μi (uαεxi )pi /α uεxi = μi |uεxi |pi uεxi have the same weak limit. In fact, suppose that
p /α
μi uαεxi + ε i uεxi → χ1
μi |uεxi |pi uεxi → χ2
∗ weakly in L∞ ,
∗ weakly in L∞ .
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
395
Let |uεxi | Ci . Define the function fi (ξ, η) ≡ (ξ α + |η|)pi /α ξ
fi : [−Ci , Ci ] × [−1, 1] → R.
For any φ ∈ L1 , we have
fi (uεxi , ε) − fi (uεxi , 0) φ dx → (χ1 − χ2 )φ dx when ε → 0.
Ω
Ω
Since the function fi is continuous, for any σ > 0 there exists δ(σ ) such that |fi (ξ, η) −
fi (ξ, 0)| σ for all ξ ∈ [−Ci , Ci ], whenever |η| δ. Therefore, for arbitrary σ > 0 and for
ε δ(σ ) we obtain
σ
f
(u
,
ε)
−
f
(u
,
0)
φ
dx
|φ| dx.
i εxi
i εxi
Ω
Ω
Thus
(χ1 − χ2 )φ dx = lim σ
f
(u
,
ε)
−
f
(u
,
0)
φ
dx
|φ| dx.
i εxi
i εxi
ε→0
Ω
Ω
Ω
Hence, χ1 = χ2 .
Proof of Theorem 2. The existence follows from Lemma 4. In fact, from Lemma 4 we have
that Φ(x) ≡ c(x)g(u) + f (x) is a bounded function. This implies the a priori estimate of the
solution in C 1+β norm (for some β ∈ (0, 1)) depending only on Φ0 and n (see, for example, [5,
Section 3.4]). The a priori estimate in C 1+β norm implies the existence of the required solution
(see, for example, [5]).
The uniqueness can be proved by standard arguments based on the maximum principle. 2
Acknowledgment
We would like to thank Professor V.N. Starovoitov for the very helpful discussion on the
monotonic operators and weak convergence.
References
[1] M. Clapp, M. del Pino, M. Musso, Multiple solutions for a non-homogeneous elliptic equation at the critical exponent, Proc. Roy. Soc. Edinburgh Sect. A 134 (1) (2004) 69–87.
[2] Q. Dai, J. Yang, Positive solutions of inhomogeneous elliptic equations with indefinite data, Nonlinear Anal. 58 (5–
6) (2004) 571–589.
[3] I. Fragala, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations,
Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (5) (2004) 715–734.
[4] H. Fujita, On the nonlinear equations u + eu = 0 and ∂v/∂t = v + ev , Bull. Amer. Math. Soc. 75 (1969) 132–
135.
[5] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer, 1983.
[6] S.N. Kruzhkov, Quasilinear parabolic equations and systems with two independent variables, Tr. Semin. im.
I.G. Petrovskogo 5 (1979) 217–272 (in Russian); English transl. in: Topics in Modern Math., Consultant Bureau,
New York, 1985.
396
Al.S. Tersenov, Ar.S. Tersenov / J. Differential Equations 235 (2007) 376–396
[7] G.M. Lieberman, Gradient estimates for anisotropic elliptic equations, Adv. Differential Equations 10 (7) (2005)
767–812.
[8] J.L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod/Gauthier–Villars,
Paris, 1969, 554 pp.
[9] S. Pokhozhaev, Eigenfunctions of the equation u + λf (u) = 0, Sov. Math. Dokl. 6 (1965) 1408–1411.
[10] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré
Anal. Non Linéaire 9 (3) (1992) 281–304.
[11] Al.S. Tersenov, On quasilinear non-uniformly elliptic equations in some non-convex domains, Comm. Partial Differential Equations 23 (11–12) (1998) 2165–2185.
[12] Al.S. Tersenov, Ar.S. Tersenov, On the Bernstein–Nagumo’s condition in the theory of nonlinear parabolic equations, J. Reine Angew. Math. 572 (2004) 197–217.
[13] Al.S. Tersenov, Dirichlet problem for a class of quasilinear elliptic equations, Math. Notes 76 (4) (2004) 546–557.